We study the recognition complexity of subgraphs of 2- and 3-connected planar cubic graphs. Recently, we presented [ESA 2022] a quadratic-time algorithm to recognize subgraphs of planar cubic bridgeless (but not necessarily connected) graphs, both in the variable and fixed embedding setting (the latter only for 2-connected inputs). Here, we extend our results in two directions: First, we present a quartic-time algorithm to recognize subgraphs of 2-connected planar cubic graphs in the fixed embedding setting, even for disconnected inputs. Second, we prove NP-hardness of recognizing subgraphs of 3-connected planar cubic graphs in the variable embedding setting.

翻译：我们研究了2-连通和3-连通平面三次图子图的识别复杂度。近期，我们在[ESA 2022]中提出了一种二次时间算法，用于识别平面三次无桥图（不一定连通）的子图，该算法适用于可变嵌入和固定嵌入设定（后者仅针对2-连通输入）。本文从两个方向扩展了我们的结果：首先，我们提出了一种四次时间算法，在固定嵌入设定下识别2-连通平面三次图的子图，即使输入图是非连通的。其次，我们证明了在可变嵌入设定下识别3-连通平面三次图子图问题是NP难的。